AI 中文总结
研究单叶函数类\(\mathcal{S}\)中函数对数系数的托普利兹行列式尖锐界，涵盖\(\alpha\)阶星形、凸、强星形、强凸及有界转向函数子类，还涉及相关逆函数对数系数的托普利兹行列式，给出主要结果及特殊情况的精确界。
AI 中文摘要
设\(\mathcal{S}\)为在开单位圆盘\(\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}\)内解析且单叶的函数\(f(z) = z + \sum_{n=2}^{\infty} a_n z^n\)的类。本文确定了其元素为\(\mathcal{S}\)中函数\(f\)的对数系数的托普利兹行列式的尖锐界。此外，研究了相关逆函数对数系数的相应托普利兹行列式。为属于\(\mathcal{S}\)的几个著名子类的函数建立了这些尖锐界，即\(\alpha\)阶星形函数类\(\mathcal{S}^*(\alpha)\)、\(\alpha\)阶凸函数类\(\mathcal{C}(\alpha)\)、\(\alpha\)阶强星形和强凸函数类\(\mathcal{S}^*_{\alpha}\)和\(\mathcal{C}_{\alpha}\)以及有界转向函数类\(\mathcal{R}(\alpha)\)。作为主要结果的特殊情况，得到了这些行列式对于星形、凸和有界转向函数的经典类别的精确界。
英文摘要
Let $\mathcal{S}$ denote the class of functions $f(z) = z + \sum_{n=2}^{\infty} a_n z^n$ that are analytic and univalent in the open unit disk $\mathbb{D} = \{z \in \mathbb{C} : |z| < 1\}$. In this paper, we determine the sharp bounds of the Toeplitz determinants whose entries are the logarithmic coefficients of $f \in \mathcal{S}$. Furthermore, we investigate the corresponding Toeplitz determinants for the logarithmic coefficients of the associated inverse functions. These sharp bounds are established for functions belonging to several well-known subclasses of $\mathcal{S}$, namely, the classes $\mathcal{S}^*(α)$ of starlike functions of order $α$, $\mathcal{C}(α)$ of convex functions of order $α$, $\mathcal{S}^*_α$ and $\mathcal{C}_α$ of strongly starlike and strongly convex functions of order $α$, and $\mathcal{R}(α)$ of functions with bounded turning. As special cases of our main results, we obtain the exact bounds of these determinants for the classical classes of starlike, convex, and bounded turning functions.
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